The aim of this master’s thesis was to give necessary and sufficient conditions for Lusin’s theorem for capacity. Lusin’s theorem is an important theorem of measure theory, but in this thesis, we replace the measure with capacity. The capacity is a function that has properties similar to the measure, but slightly weaker. In the introductory part, we define the concept of the capacity, present some of its properties and give some examples of capacities. After that, we prove several results from mathematical analysis, metric spaces and measure theory that we need for the proof of the main result. Also, we define an important concept of regular capacity and show in what relation is the regularity with the continuity from above on the sequences of open sets. In the main part of the thesis, we prove that the continuity from above on the sequences of open sets is the crucial property that subadditive capacity on the compact metric space must possess for Lusin’s theorem to hold true for this class of capacities. In fact, we show even more, that is, we show that it is a necessary and sufficient condition for Lusin’s theorem. Lastly, we show two consequences of Lusin’s theorem for capacity concerning the boundness of measurable functions and approximation of an arbitrary measurable function with a continuous function.